By letting \(W_0^k:=V_0^k\) , we obtain a hierarchical decomposition \(V_n^k=\bigoplus _{0 \le l \le n} W_l^k\) , i.e., MRA of space \(V_n^k\) .
A set of orthonormal basis can be defined on \(W_l^k\)  as follows. When \(l=0\) , the basis \(v^0_{i,0}(x)\) , \( i=0,\ldots ,k\)  are the normalized shifted Legendre polynomials in \([0,1]\) . When \(l>0\) , we will use the Alpert's orthonormal multiwavelets [1]} as the bases, which have been employed to develop a class of sparse grid DG methods for solving PDEs in high dimensions [2]}, [3]}, [4]}, [5]}. In this paper, we adopt the notation
\(v^j_{i,l}(x),\quad i=0,\ldots ,k,\quad j=0,\ldots ,2^{l-1}-1.\) 
